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Properties

Label 2-115-23.13-c3-0-15
Degree 22
Conductor 115115
Sign 0.952−0.304i0.952 - 0.304i
Analytic cond. 6.785216.78521
Root an. cond. 2.604842.60484
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.784 + 0.905i)2-s + (6.23 + 4.00i)3-s + (0.934 − 6.49i)4-s + (2.07 − 4.54i)5-s + (1.26 + 8.79i)6-s + (8.12 + 2.38i)7-s + (14.6 − 9.43i)8-s + (11.6 + 25.4i)9-s + (5.75 − 1.68i)10-s + (13.9 − 16.1i)11-s + (31.8 − 36.7i)12-s + (−12.7 + 3.74i)13-s + (4.21 + 9.23i)14-s + (31.1 − 20.0i)15-s + (−30.3 − 8.89i)16-s + (3.21 + 22.3i)17-s + ⋯
L(s)  = 1  + (0.277 + 0.320i)2-s + (1.20 + 0.771i)3-s + (0.116 − 0.812i)4-s + (0.185 − 0.406i)5-s + (0.0860 + 0.598i)6-s + (0.438 + 0.128i)7-s + (0.649 − 0.417i)8-s + (0.430 + 0.941i)9-s + (0.181 − 0.0533i)10-s + (0.382 − 0.441i)11-s + (0.766 − 0.884i)12-s + (−0.272 + 0.0799i)13-s + (0.0805 + 0.176i)14-s + (0.536 − 0.344i)15-s + (−0.473 − 0.139i)16-s + (0.0458 + 0.318i)17-s + ⋯

Functional equation

Λ(s)=(115s/2 ΓC(s) L(s)=((0.952−0.304i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(115s/2 ΓC(s+3/2) L(s)=((0.952−0.304i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 115115    =    5⋅235 \cdot 23
Sign: 0.952−0.304i0.952 - 0.304i
Analytic conductor: 6.785216.78521
Root analytic conductor: 2.604842.60484
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ115(36,⋅)\chi_{115} (36, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 115, ( :3/2), 0.952−0.304i)(2,\ 115,\ (\ :3/2),\ 0.952 - 0.304i)

Particular Values

L(2)L(2) ≈\approx 2.84234+0.443475i2.84234 + 0.443475i
L(12)L(\frac12) ≈\approx 2.84234+0.443475i2.84234 + 0.443475i
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1+(−2.07+4.54i)T 1 + (-2.07 + 4.54i)T
23 1+(109.+14.3i)T 1 + (109. + 14.3i)T
good2 1+(−0.784−0.905i)T+(−1.13+7.91i)T2 1 + (-0.784 - 0.905i)T + (-1.13 + 7.91i)T^{2}
3 1+(−6.23−4.00i)T+(11.2+24.5i)T2 1 + (-6.23 - 4.00i)T + (11.2 + 24.5i)T^{2}
7 1+(−8.12−2.38i)T+(288.+185.i)T2 1 + (-8.12 - 2.38i)T + (288. + 185. i)T^{2}
11 1+(−13.9+16.1i)T+(−189.−1.31e3i)T2 1 + (-13.9 + 16.1i)T + (-189. - 1.31e3i)T^{2}
13 1+(12.7−3.74i)T+(1.84e3−1.18e3i)T2 1 + (12.7 - 3.74i)T + (1.84e3 - 1.18e3i)T^{2}
17 1+(−3.21−22.3i)T+(−4.71e3+1.38e3i)T2 1 + (-3.21 - 22.3i)T + (-4.71e3 + 1.38e3i)T^{2}
19 1+(16.8−117.i)T+(−6.58e3−1.93e3i)T2 1 + (16.8 - 117. i)T + (-6.58e3 - 1.93e3i)T^{2}
29 1+(−39.3−273.i)T+(−2.34e4+6.87e3i)T2 1 + (-39.3 - 273. i)T + (-2.34e4 + 6.87e3i)T^{2}
31 1+(−53.4+34.3i)T+(1.23e4−2.70e4i)T2 1 + (-53.4 + 34.3i)T + (1.23e4 - 2.70e4i)T^{2}
37 1+(81.0+177.i)T+(−3.31e4+3.82e4i)T2 1 + (81.0 + 177. i)T + (-3.31e4 + 3.82e4i)T^{2}
41 1+(−11.4+25.1i)T+(−4.51e4−5.20e4i)T2 1 + (-11.4 + 25.1i)T + (-4.51e4 - 5.20e4i)T^{2}
43 1+(−24.8−15.9i)T+(3.30e4+7.23e4i)T2 1 + (-24.8 - 15.9i)T + (3.30e4 + 7.23e4i)T^{2}
47 1+313.T+1.03e5T2 1 + 313.T + 1.03e5T^{2}
53 1+(−22.8−6.70i)T+(1.25e5+8.04e4i)T2 1 + (-22.8 - 6.70i)T + (1.25e5 + 8.04e4i)T^{2}
59 1+(630.−185.i)T+(1.72e5−1.11e5i)T2 1 + (630. - 185. i)T + (1.72e5 - 1.11e5i)T^{2}
61 1+(−379.+243.i)T+(9.42e4−2.06e5i)T2 1 + (-379. + 243. i)T + (9.42e4 - 2.06e5i)T^{2}
67 1+(−187.−216.i)T+(−4.28e4+2.97e5i)T2 1 + (-187. - 216. i)T + (-4.28e4 + 2.97e5i)T^{2}
71 1+(462.+533.i)T+(−5.09e4+3.54e5i)T2 1 + (462. + 533. i)T + (-5.09e4 + 3.54e5i)T^{2}
73 1+(−112.+784.i)T+(−3.73e5−1.09e5i)T2 1 + (-112. + 784. i)T + (-3.73e5 - 1.09e5i)T^{2}
79 1+(−254.+74.7i)T+(4.14e5−2.66e5i)T2 1 + (-254. + 74.7i)T + (4.14e5 - 2.66e5i)T^{2}
83 1+(−167.−365.i)T+(−3.74e5+4.32e5i)T2 1 + (-167. - 365. i)T + (-3.74e5 + 4.32e5i)T^{2}
89 1+(−1.26e3−815.i)T+(2.92e5+6.41e5i)T2 1 + (-1.26e3 - 815. i)T + (2.92e5 + 6.41e5i)T^{2}
97 1+(136.−299.i)T+(−5.97e5−6.89e5i)T2 1 + (136. - 299. i)T + (-5.97e5 - 6.89e5i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−13.64050676075465831173950061772, −12.23717532257592046267448630128, −10.70674973102399314484124540965, −9.854527789619740706090083119555, −8.904680503401845024876378781599, −7.918078682602477205144535858419, −6.21181770312597080269128240910, −4.93977740622743804564781407982, −3.72109179904234241957900614886, −1.79135031696026705708514633735, 2.01562685224140303758394294204, 2.99899630230164391867461941914, 4.47241533386485960897914041420, 6.73859472670700056215255429899, 7.66184243752908981877913445407, 8.449894525182871111440121940118, 9.711437092084506643121389817959, 11.25870616240833839353635000844, 12.15413476319451951064088255185, 13.25791919891705288978421778582

Graph of the ZZ-function along the critical line